A student's guide to numerical methods
By: Ian H. Hutchinson, Massachusetts Institute of Technology
Published: 2015
Language: eng
Number of pages: 207
ISBN-13: 9781107095670; 9781107479500
Subjects: Mathematics; Numerical analysis; Study and teaching; Measurement; Number concept; Matrices
Description: This concise, plain-language guide for senior undergraduates and graduate students aims to develop intuition, practical skills and an understanding of the framework of numerical methods for the physical sciences and engineering. It provides accessible self-contained explanations of mathematical principles, avoiding intimidating formal proofs. Worked examples and targeted exercises enable the student to master the realities of using numerical techniques for common needs such as solution of ordinary and partial differential equations, fitting experimental data, and simulation using particle and Monte Carlo methods. Topics are carefully selected and structured to build understanding, and illustrate key principles such as: accuracy, stability, order of convergence, iterative refinement, and computational effort estimation. Enrichment sections and in-depth footnotes form a springboard to more advanced material and provide additional background. Whether used for self-study, or as the basis of an accelerated introductory class, this compact textbook provides a thorough grounding in computational physics and engineering.
Notes: Includes bibliographical references and index.
Table of contents: title: pt. I. Fitting functions to data. Exact fitting ; Approximate fitting ; Tomographic image reconstruction ; Efficiency and non - linearity; title: pt. II. Ordinary differential equations. Reduction to first order ; Numerical integration of initial - value problems ; Multidimensional stiff equations : implicit schemes ; Leap - frog schemes; title: pt. III. Two - point boundary conditions. Examples of two - point problems ; Shooting ; Direct solution ; Conservative differences, finite volumes; title: pt. IV. Partial differential equations. Examples of partial differential equations ; Classification of partial differential equations ; Finite - difference partial derivatives; title: pt. V. Diffusion. parabolic partial differential equations. Diffusion equations ; Time - advance choices and stability ; Implicit advancing matrix method ; Multiple space dimensions ; Estimating computational cost; title: pt. VI. Elliptic problems and iterative matrix solution. Elliptic equations and matrix inversion ; Convergence rate ; Successive over - relaxation ; Iteration and non - linear equations; title: pt. VII. Fluid dynamics and hyperbolic equations. The fluid momentum equation ; Hyperbolic equations ; Finite differences and stability; title: pt. VIII. Boltzmann's equation and its solution. The distribution function ; Conservation of particles in phase - space ; Solving the hyperbolic Boltzmann equation ; Collision term; title: pt. IX. Energy - resolved diffusive transport. Collisions of neutrons ; Reduction to multigroup diffusion equations ; Numerical representation of multigroup equations; title: pt. X. Atomistic and particle - in - cell simulation. Atomistic simulation ; Particle - in - cell codes; title: pt. XI. Monte Carlo techniques. Probability and statistics ; Computational random selection ; Flux integration and injection choice; title: pt. XII. Monte Carlo radiation transport. Transport and collisions ; Tracking, tallying, and statistical uncertainty; title: pt. XIII. Next steps. Finite - element methods ; Discrete Fourier analysis and spectral methods ; Sparse - matrix iterative Krylov solution ; Fluid evolution schemes; title: Appendix A. Summary of matrix algebra. Vector and matrix multiplication ; Determinants ; Inverses ; Eigenanalysis.